Associative spectra of graph algebras II

A necessary and sufficient condition is presented for a graph algebra to satisfy a bracketing identity. The associative spectrum of an arbitrary graph algebra is shown to be either constant or exponentially growing.

Perfect matchings, rank of connection tensors and graph homomorphisms

We develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.

On band modules and $$\tau $$-tilting finiteness

In this paper, motivated by a $$\tau $$ τ -tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $$\tau $$ τ -tilting finite if and only if no band module is a brick. We also recover a criterion for the $$\tau $$ τ -tilting finiteness of Brauer graph algebras in terms of the Brauer graph.

Associative spectra of graph algebras I

Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.

Algebraic Operations on Spatiotemporal Data Based on RDF

In the context of the Semantic Web, the Resource Description Framework (RDF), a language proposed by W3C, has been used for conceptual description, data modeling, and data querying. The algebraic approach has been proven to be an effective way to process queries, and algebraic operations in RDF have been investigated extensively. However, the study of spatiotemporal RDF algebra has just started and still needs further attention. This paper aims to explore an algebraic operational framework to represent the content of spatiotemporal data and support RDF graphs. To accomplish our study, we defined a spatiotemporal data model based on RDF. On this basis, the spatiotemporal semantics and the spatiotemporal algebraic operations were investigated. We defined five types of graph algebras, and, in particular, the filter operation can filter the spatiotemporal graphs using a graph pattern. Besides this, we put forward a spatiotemporal RDF syntax specification to help users browse, query, and reason with spatiotemporal RDF graphs. The syntax specification illustrates the filter rules, which contribute to capturing the spatiotemporal RDF semantics and provide a number of advanced functions for building data queries.